Is invisibility a synonym for non-existence?

No necessarily, Jehovah God, his only begotten son & the angels are real, and yet you and I can not see them, but the heavenly creatures can not been seen by mankind, but these others like Satan the Devil can not been seen except by God & his creatures.

"Praise Jah, you people!"

Like my invisible character for the Q&A is invisible! How cool is that?

God & His invisible angels ; even Satan the Devil can't be seen unless God & the other creatures like Jesus Christ, these can be seen, according to God's word!

"Praise Jah, you people!"

No, it is a synonym with not visible, that's to say something that either does not interact with light or that it interacts with it without affecting its normal course.

Some example of invisible things that exist : sound, gravity, electricity, etc

Be careful, because something is invisible doesn't mean it doesn't interact with things that are visible. Thus we are able to perceive the effects of those things through their interactions. For example, you do not see sound, but you can hear it, feel it and see things it moves. You do not see gravity but you see things that are pulled by it. You do not see electricity but you can see your computer and lights work, you can see the ionized air when there's lightning, etc.

Now, sometimes invisibility is used as a quasi synonym of undetectable. Something really undetectable cannot be proven to exist because it doesn't interact with anything.

No, because you cannot have something from nothing.

All (astro) physicists agree that all matter is within the universe, therefore whatever caused the universe had to be immaterial since it existed "outside" and "before" the universe, so to speak.

Immaterial = invisible (not necessarily but for the sake of the question I think you're asking) if you're confused.

Invisibility is a synonym for being ignored and unnoticed.

No. Mr. Dictionary is your friend even for the religion and spirituality category.

Certainly things that are immaterial, beyond space, beyond time, outside of reality, etc... can be rightly understood as non-existent. If your describing something that has nothing to do with reality and cannot exist within reality it is because it is not real.

No

In the case of the God of the Bible, yes.

Nope. That means it's unable to be seen. You know how you can't see air but it still exists? Different thing.